Johan Hombert† David Thesmar‡
Abstract
We present a model where arbitrageurs operate on an asset market
that can be hit by information shocks. Before entering the market,
arbitrageurs are allowed to optimize their capital structure, in
order to take advantage of potential underpricing. We find that, at
equilibrium, some arbitrageurs always receive funding, even in low
information environ- ments. Other arbitrageurs only receive funding
in high information environments. The model makes two easily
testable predictions: first, arbitrageurs with stable funding
should experience more mean reversion in returns, in particular
following low performance. Sec- ond, this larger mean reversion
should be lower, if many other funds have stable funding. We test
these predictions on a sample of hedge funds, some of which impose
impediments to withdrawal to their investors.
1 Introduction
In the literature on limits to arbitrage, a widening of the
mispricing of an asset may lead arbitrageurs to unwind their
positions, which further amplifies the initial mispricing (Shleifer
and Vishny, 1997, Gromb and Vayanos, 2002). Such forced unwinding
occurs because, as arbitrageurs lose money on their trades, their
investors (brokers, banks, limited partners etc) demand early
reimbursement of their claims. Thus, existing theories of the
limits to arbitrage assume that arbitrageurs cannot design their
capital structure ex ante (for instance, by taking on long term
debt) in order to avoid such value destroying events.
This paper starts from the simple fact that this assumption does
not always hold in reality, and investigates its theoretical and
empirical consequences. In the hedge fund industry, investors often
agree to limit their ability to withdraw their funds. About 20% of
the hedge funds in our sample have lockup periods of typically one,
or even two years, during which investors cannot redeem their
shares (Aragon, 2007, has a similar proportion). Once they are able
to do so, they must give the fund advance notice (typically a
month) and then obtain ∗This paper has benefited from financial
support from the BNP Paribas Hedge Fund Centre at HEC, and
the Paul Woolley Centre for the Study of Market Dysfunctionality.
We are grateful to Serge Darolles, Vincent
Pouderoux, Isabelle Serot, Guillaume Simon at SGAM-AI for decisive
help with hedge fund data. †ENSAE-CREST ‡HEC, CEPR and ECGI
1
redemption at fixed dates (typically a quarter). For the average
hedge fund in our sample, we estimate the minimum duration of funds
to be equal to 5 months, and 10 months for funds with lockup
periods. Interestingly, such share restrictions can be found with
hedge funds investing in illiquid securities (such as fixed
income), but also with funds dealing with stocks (such as “long
short equity” funds). They are what we call “limits of limits of
arbitrage”: thanks to them, some market participants can afford to
underperform in the short run while they hold on to ultimately
profitable arbitrage opportunities.
Thus, at least some arbitrageurs choose the maturity of their
investors’ claims. To under- stand the determinants and
consequences of such a capital structure decision, we first build a
model where arbitrageurs optimally design the securities that they
issue, and then engage in arbitrage on the same market.
Arbitrageurs differ in skill. We posit that arbitrageur skill
affects long term asset payoffs in some states of nature only (“low
information states”). We first find that, at equilibrium, prices in
low information states are lower, because scarce arbitraging skills
are needed to trade in these states. Furthermore, at equilibrium
investors guarantee funding to skilled arbitrageurs in low
information (low price) states, while unskilled arbitrageurs only
receive funding in high information (high price) states. The
intuition comes from the asset price equilibrium: if investors did
not guarantee funds to some arbitrageurs in low information states,
asset prices in these states would collapse, which would make
invest- ment attractive. Even when arbitrageur skill is not
contractible upon, the equilibrium capital structure choice is
separating: skilled arbitrageurs choose guaranteed funding, while
unskilled arbitrageurs choose funding only contingent on high
information (high prices). Thus, there is optimal differentiation
at equilibrium.
Our model generates two easily testable predictions. First,
conditional on past bad per- formance, funds with guaranteed
funding outperform other funds. As argued above, in low information
states, scarce skills are needed which lowers current prices. Thus,
funds with guaranteed funding invest more often in states where the
assets are underpriced, and thus outperform other funds who take
less advantage of underpricing. Our second prediction is cross
sectional: in industries where the fraction of arbitrageurs with
guaranteed funding is large, these arbitrageurs overperform other
funds less. The mechanism is deeply rooted in the model: if more
arbitrageurs receive guaranteed funding, underpricing in the low
information state will be reduced. Since these arbitrageurs benefit
from underpricing, their overperfor- mance will be reduced.
We then test these two predictions on hedge fund data. We use the
fact that, in our data, funds with impediments to withdrawal (such
as long redemption periods, or lockup periods) experience less
outflows when they underperform (Ding et al., 2008, find similar
evidence). Thus, these share restrictions, which we can observe
from the data, are a good proxy for “guaranteed funding” in our
model. We find that the first prediction of our model holds in the
data: conditional on bad past performance, funds with impediments
to withdrawal do “bounce back more”, i.e., have higher expected
returns. We also find some support for the second prediction. To
test it, we look at investment styles where impediments to
withdrawal are prevalent. We find that, in these styles, funds with
such share restrictions overperform
2
other funds relatively less than in styles where such impediments
are relatively rare, although these results are somewhat less
robust.
This paper contributes to two strands of literature. First, we
extend Shleifer and Vishny’s model of limits of arbitrage by
allowing arbitrageurs to optimally choose their capital struc- ture
in order to avoid inefficient liquidation. In this sense, our paper
is closely related to independent work by Stein (2009), which is
the only paper, to the best of our knowledge, that explicitly seeks
to endogeneize arbitrageurs’ capital structures. Compared to his
model, our theory endogeneizes the cost of external finance more
explicitly and makes testable predictions on observed arbitrageur
returns, that we can bring to the data. What is common to the two
models is that underpricing in bad states of nature leads to more
investment in these states, through the optimal structure choice.
This feedback mechanism is not present in Shleifer and Vishny
(1997) nor in Stein (2005), who does not endogeneize prices. The
rest of the limits to arbitrage literature considers the
destabilizing feedback that goes through the wealth of arbi-
trageurs. Gromb and Vayanos (2002), Brunnermeier and Pedersen
(2008) and Acharya and Viswanathan (2007) model intermediaries that
need to unwind their positions when collateral prices decrease,
which amplifies price drops. In all these models, even if
mispricing can be very large, there is no contractual way to take
advantage of this.1
Second, we shed new light on existing evidence from the (mostly
hedge) funds literature. First, our findings are related to a
recent paper on open end mutual funds by Coval and Stafford (2008):
they look at asset fire sales following massive redemptions at
mutual funds, and find a significant price impact. Thus, their
paper suggests (but does not test) that mutual fund performance
should display some persistence, in particular conditional on past
low performance. We propose a theory of why some funds may seek
protection against massive redemptions, and what returns dynamics
should look like in protected and unprotected funds. Second, some
papers show that the presence of impediments to withdrawal is
correlated with unconditional fund performance (Aragon, 2007,
Agarwal et al., 2008): their explanation is that investors earn a
premium for the illiquidity of their investment. Other papers have
informally argued that hedge funds act as liquidity providers (see,
e.g., Agarwal, Fung, Loon and Naik, 2008). The present article
suggests a potential reason why illiquid funds can afford to issue
illiquid shares in the first place: because illiquidity allows them
to reap the gains of arbitrage, they can pay the illiquidity
premium to their investors. Third, we develop and test a theory of
the mean reversion of fund returns. Interestingly, some of the
existing hedge funds literature has focused on the positive
relation between autocorrelation and share
1Also related to this paper is Lerner and Schoar (2004). They test
a model where (private equity) fund
managers make their shares illiquid in order to select “patient”
investors. Their capital structure focus differs
from ours in an important way: they look at the ability to sell
shares to other investors, while we look at
the ability to sell shares to the fund. In addition, if we were to
transpose their mechanism in our setting
where funds interact through buying and selling the same asset,
funds would screen “patient investors” until
mispricing disappears. There would be no prediction on the link
between share restriction and fund returns
dynamics.
Casamatta and Pouget (2008) solve a model where investors give fund
managers incentives to search for
information on assets. In their model, the optimal contract
features short term performance pay. The cost of
such contracts is that they reduce market efficiency by deterring
information acquisition.
3
restrictions (see, e.g., Aragon, 2007) while we find clear evidence
of such a negative relation. The difference between these studies
and ours is the frequency at which autocorrelation is computed: we
work at the annual level, while existing papers work at the monthly
level. At the monthly frequency, the existing literature argues
that reported returns of illiquid assets are smoothed. At the
annual frequency, this paper argues that arbitrage induces a mean
reversion in fund returns. To some extent, such evidence is
reminiscent to insights from the strategic allocation literature
(Campbell and Viceira, 2002) which argues that long term investors
have a comparative advantage at investing in mean reverting
assets.
The rest of the paper follows a simple structure. Section 2
describes, solves the model and derives predictions and comparative
statics. Section 3 tests the model. Section 4 concludes.
2 Model
2.1 Set-Up
There are competitive, risk neutral, investors. Investors want to
purchase an asset which is in unit supply, but cannot do so
themselves. They delegate this task to a measure 1 of fund
managers. Fund managers are risk neutral and limitedly liable; each
of them starts with initial wealth A.
2.1.1 Sequence of Events
There are four periods t = 0, 1, 2, 3 and the discount rate is
zero. At t = 0, investors contract with managers. The optimal
contract will specify the amount of funds that the investor will
entrust to the manager, both in t = 1 and 2, and conditional on the
state of nature in t = 2.
At date t = 1, each fund manager learns about the asset he will be
trading: the acquired knowledge will only be useful in period t =
2. Learning effort costs C to the manager. As explained below, we
assume that learning effort is not contractible. With high learning
effort, the manager becomes skilled with probability µ; with low
learning effort, with probability µ−µ. The manager does not know
whether he is skilled until period 3. After the learning phase,
managers use entrusted funds to purchase assets at unit price P .
The market for assets clears.
At date t = 2, the market can be in one of three states. With
probability λU , the market is in state U : in this state,
knowledge acquired in period 1 is useless (think for instance of a
bull market where everyone can generate high returns). It becomes
public knowledge that the asset will generate t = 3 cash flows of V
> 0. All fund managers liquidate their positions from t = 1, pay
off their investors, and use newly entrusted funds, as specified in
the optimal contract, to repurchase the same assets. The market
clears again at price PU .
With probability λM , the market is in state M . In this state, we
assume that a second asset, which is a priori not distinguishable
from the first asset, appears. Because they cannot be
differentiated from each other, both assets trade at the same
price, but we assume that the second asset has zero present value,
while the first asset has, as in state U , a PV of V .
4
Furthermore, we assume that only a fraction µ of the managers picks
the “right” asset. The important hypothesis is that, in state M ,
the asset PV does not depend on t = 1 effort. Thus, compared to
state U , state M is a bad state, in the sense that there is less
information than in state U , but the state is equally bad for all
managers, irrespective of their t = 1 learning decisions. In this
state, the market for assets clears at price PM .
Last, with probability λD, the market is in state D. Exactly as in
state M , a second asset appears that has a PV of zero, but this
time skilled managers can differentiate between the two. Thus, an
important difference between states M and D is that in state D,
date 1 learning effort matters. In this sense, state D also is a
bad state, but it is worse for managers who did not learn in t = 1.
In this state, the “right” asset market clears at price PD, which
is also the price of the “wrong” asset whose market we do not
model.
At date t = 3, assets held in portfolios mature and payoffs are
realized. If the “right” asset is held, its payoff is V . In states
M and D, we assume that only V −B can be pledged to the investors.
We think of B as the rent of an unmodeled agency conflict in period
3: for instance, the manager can sell the asset on a black market
for price B and consume the proceeds. To simplify exposition, we
assume that this agency conflict does not exist in state U , in
which case the entire present value of the asset V can be pledged
to investors. All intuitions of this model would carry through
without this assumption.
All in all, states U , M and D vary along two key dimensions.
First, in state U , expected cash flows from assets are higher than
in states M and D. Expected present value in U is always V ; in
state M it is only µV . In state D only skilled managers will be
able to buy good assets, so the expected payoff is at most µV .
This feature of the model (µ < 1) is not entirely necessary for
most intuitions to carry through; we will explain why later on. The
second difference between the three states is that, in state D,
managerial skill matters. Thus, a manager who is committed more
funds in state D will have more incentive to learn. This second
dimension of our model is essential.
2.1.2 Contracts
We assume that the financial contract specifies four amounts
entrusted to the manager: I, in period 1, and (IU , IM , ID) in
period 2, conditional on states U , M and D. Thus, we make two
implicit, and mostly simplifying, assumptions. First, we assume
that date 1 learning effort cannot be contracted upon. As we
explain later on, this assumption is not necessary to obtain our
results. Second, we assume that the date 2 state of nature is
contractible. This is the case for instance if period 2 returns are
contractible.2 It is precisely the goal of this paper to study the
impact of the contingent financing of arbitrageurs.
The financial contract could also include a compensation for the
manager in some states of nature, in order to induce the manager to
put in high learning effort. To simplify the analysis, we will
impose restrictions on parameter values such that the incentive
compatibility
2Indeed, the asset prices in the three period 2 states of nature
will be different from each others. Therefore,
the rational expectations equilibrium we will find when the state
of nature is contractible, persists when the
state of nature is observable but not verifiable, and returns are
contractible.
5
constraint is not binding. Therefore, adding an incentive fee on
top of the private benefit B per unit of asset will not be part of
an optimal contract.
In the empirical application, we will think of contracts where ID
> 0 as contracts with impediments to withdrawals. Such contracts
guarantee t = 2 inflows even when the fund underperforms, i.e.,
when asset prices go down in state D.
2.1.3 Modeling Strategy
Before we solve the model, it is worthwhile to discuss our
modelling strategy. With moral hazard, another possible strategy
could have been to think of withdrawals as an ex ante opti- mal
“punishment” strategy. Underperforming managers are punished for
low effort provision, while well performing managers are rewarded
through continuation. Such a model delivers similar comparative
static properties as the one we study in this paper, but the
implied con- tract has the important drawback of not being
renegotiation-proof. Once low effort has been provided, assets are
fairly priced in equilibrium, and their expected return is non
negative. Thus, shutting down the fund is never an optimal decision
ex post and the punishment is non credible.
One second alternative would have been to model limits of arbitrage
as arising because in- vestors learn about the fund manager’s
skill, assuming such a skill is fixed from the beginning, i.e., not
obtained through learning. If the fund underperforms, then it
becomes optimal to withdraw investment because the chances that the
manager is incompetent are high. In such a model, there would be no
reason for an investor to lock his money up in the fund because
there is no efficiency gain to do so.
To make impediments to withdrawals optimal from a contracting
perspective, they must have an incentive property, so we choose a
moral hazard setting: learning entails an “effort”. An alternative
model would be to assume that managers have fixed types (skilled or
unskilled) and that investors seek to design separating contracts.
Such a model would be almost identical to the model we present in
this paper, except that learning is exogenous. Such a model would
generate identical predictions to the ones we derive and test
here.
2.2 Solving the Model
We first derive the optimal contracts for given expected asset
prices, and then solve for the rational expectations equilibrium of
the asset market. This allows us to (1) characterize the
equilibrium and (2) find a relationship between impediments to
withdrawals (ID > 0), and the equilibrium returns of the
funds.
2.2.1 Optimal Contracts
In this section, we take the sequence of future prices P , PU , PM
and PD as given. The optimal contract solves the manager’s
objective function, which is the project’s NPV, under the
constraints that the profit pledgeable to investors is nonnegative
and that the manager exerts the desired level of learning effort
(Tirole, 2006).
6
max I, IU , IM , ID
I[λUPU + λMPM + λDPD − P ] + λUIU [V − PU ]
+λMIM [µV − PM ] + λDID[µV − PD],
subject to
+λMIM [µ(V −B)− PM ] + λDID[µ(V −B)− PD] +A ≥ 0,
and λDµBID > C.
The objective function is the overall NPV of the fund. The first
term is the total profit made between period 1 and 2, which is
equal to the expected price increase times the amount invested in t
= 1. Given that this profit is free from any agency consideration,
it can be pledged to the investor at 100%, which is why it also
appears as such in the first (investor participation) constraint.
The second term is the t = 2 NPV realized in state U , which can
also be fully pledged. The third term is the expected NPV in state
M . In this case, the manager will purchase the right asset with
probability µ, since learning effort has been made, and, as appears
in the first constraint, only µ(V −B) per asset purchased can be
pledged at t = 0. In state D, the conditional expected payoff per
asset is the same, because the manager puts in high effort. The
second constraint is the manager’s incentive compatibility
constraint which ensures that, in period 1, high learning effort is
always preferred. Given our parameters restrictions below, this
constraint will never bind at equilibrium.
It is clear from the above problem that
P = λUPU + λMPM + λDPD
will have to hold in equilibrium. If this is not the case, I will
be equal to +∞ or −∞. Hence, markets in t = 1 are fully efficient
in this model because there is no agency friction in t = 1: any
profit from arbitrage is pledgeable, so that infinite amount of
funds can be used to finance arbitrageurs. This reduces arbitrage
opportunities to zero. For the same reason, the same happens in
state U : PU = V . Thus, all funds receive an indeterminate amount
of funding in state U .
Moreover, it is easy to see that PM ≤ µV and PD ≤ µV have to hold
in equilibrium, otherwise no fund would be willing to hold the
asset in state M or in state D. At the same time, PM > µ (V −B)
and PD > µ (V −B). This comes from the fact that the marginal
pledgeable income of investment has to be strictly negative in
equilibrium. If this is not the case, fund managers could raise
money to invest more, as the NPV of doing so is strictly positive.
This would contradict the equilibrium.
7
Given these properties and the convenient linearity of the problem,
we obtain that
ID = 1 λD
PD − µ(V −B) (µV − PD)− C,
if PD < PM . Hence, if the price in state D is low enough
compared to the price in state M , it is then optimal to allocate
all pledgeable income in state D where the asset is relatively
cheap. In contrast, as soon as PM < PD,
ID = 0,
PM − µ(V −B) (µV − PM )− C.
When the asset is cheap in state M , it is optimal to allocate all
pledgeable income in state M .
Computations and expressions are very similar when the learning
effort is low. In this case, the fund will invest in state D only
when PM > µ
µ−µPD, and in state M only when the reverse inequality holds. This
leads us to the following lemma:
Lemma 1. For given asset prices PM ≤ V and PD ≤ µV , there are ve
regimes: In all regimes, all funds receive funding in state U . In
addition,
1. PM < PD: both high and low eort funds invest in state M
only.
2. PM = PD: high eort funds are indierent between investing in
state M and D; low eort funds invest in state M only.
3. PD < PM < µ µ−µPD: high eort funds invest in state D only;
low eort funds invest
in state M only.
4. PM = µ µ−µPD: high eort funds invest in state D only; low eort
funds are indierent
between investing in state M and D.
5. µ µ−µPD < PM : both high and low eort funds invest in state D
only.
The results of this lemma are intuitive: high PM discourages funds
to invest in state M . In addition, high effort funds have higher
returns to investing in state D, since this is when learning effort
pays off. Hence, high effort funds are ready to invest in state D
for higher levels of PD.
The above lemma also indicates that cases 1 and 5 cannot be
equilibrium outcomes, since in these cases there is no demand for
assets in either state D or M . Putting aside the knife- edge cases
2 and 4, this suggests that in equilibrium both levels of learning
effort coexist: high
8
effort funds invest in state D only, while low effort funds invest
in state M only.3 We now turn to the description of the
equilibrium.
2.2.2 Equilibrium
Following the above discussion, we restrict ourselves to PD < PM
< µ µ−µPD. Let α be
the equilibrium fraction of high effort funds. In equilibrium,
since both categories of funds coexist, funds have to be
indifferent, ex ante, between putting in high effort (and buy in
state D) or low effort (and buy in state M):
µV − PD PD − µ (V −B)
− C
A =
. (1)
We now need to compute equilibrium prices PM and PD. Aggregate
asset demand by funds in state M and state D has to be equal to
supply (assumed equal to 1). Hence
PM = µ (V −B) + µ(1− α)A
λM , (2)
λD . (3)
The price in each state is higher, the higher the expected payoff,
the higher the equity of managers, and the higher the number of
funds operating in this state. Plugging back (2) and (3) into
indifference condition (1), we obtain the following equation for
the equilibrium α:
λD α − C
. (4)
It is straightforward to see that α ∈ (0; 1). Moreover, α is
increasing in λD, and decreasing in λM and C/B. When the cost of
making effort (C) decreases, or when the gains of making effort
(λD) are larger, there will be more high effort funds operating in
equilibrium.
So far we have assumed that, once the contract is signed, the fund
manager puts in the expected amount of effort. It is
straightforward to see that a manager with ID = 0 will make no
learning effort, since it will never pay off. A fund manager with
ID > 0 puts in high effort if and only if his incentive
constraint is satisfied
ID = 1 αµ
λDBµ , (5)
i.e., ID is large enough to make the gain of learning λDµBID larger
than the effort cost C. Condition (5) ensures that providing the
manager with an incentive fee on top of the private benefit B is
not part of an optimal contract.
3Case 2 cannot actually arise in equilibrium, otherwise a high
training effort fund would be indifferent in
t = 2 between investing in state M or D. In t = 1, it would then be
optimal to make low effort and invest in
state M only, to save the training effort cost, hence there would
be not demand for the asset in state M . By
contrast, case 4 can be an equilibrium outcome for some parameter
values. To clarify exposition, we rule them
out in the following.
9
Finally, we need to ensure that asset prices in period 2 are below
their fundamental value in equilibrium, else conditions (2) and (3)
do not apply. This occurs if and only if
A < λMB
1− α . (6)
Intuitively, if fund managers have little equity, their demand will
be so low that prices do not reach their fundamental values, even
in state M .
Hence, an equilibrium is defined by equation (4), under conditions
(5) and (6). Equilibrium prices also have to satisfy PD < PM
< µ
µ−µPD. The following proposition characterises such an equilibrium,
and provides a parameter condition for its existence.
Proposition 1. There exist A and µ < µ such that, if
µ > µ and A < A,
1. The only equilibrium is an equilibrium where α ∈ (0; 1) funds
make high learning eort and are only committed funds in states U
and D, and 1 − α funds make low learning eort and are only
entrusted funds in states U and M .
2. α is dened by equation (4). Equilibrium prices are such that PD
< PM .
3. The ex ante optimal contract is renegotiation-proof in
equilibrium.
Proof. Let α be the (unique) positive solution of equation (4). Let
A = λMB/(1 − α). The condition A < A is equivalent to condition
(6) which is therefore satisfied. Let µ = Cαµ/λDB. The condition µ
> µ ensures that the incentive compatibility constraint for high
effort funds holds. From (4), it is easy to see that µ <
µ.
From equilibrium prices (2) and (3), it is easy to obtain
that
PM − PD = (
is strictly positive by virtue of (4). Finally, straightforward
manipulations show that PM < µ
µ−µPD is equivalent to
µ µ
> 1− α
by equation (4), which holds when µ > µ.
The optimal contract is renegotiation-proof because, in
equilibrium, the asset price is always above the marginal
pledgeable payoff, but below the marginal NPV. As a result, the
manager cannot raise new funds, since he can only promise a
negative income V − B − Pi, i = M,D, nor is he willing to cut down
investment, since he obtains utility B per asset invested. Put
differently, the contract is renegotiation-proof because
continuation is as optimal
10
ex post as it is ex ante: the size of the surplus does not increase
nor decrease and there is thus no scope for renegotiation.
One can compute the relative underpricing in state D as the
difference PM − PD. Given that the expected PV of the assets is µ
(V −B) in both states, PM − PD measures the price difference that
is not due to a difference in expected payoff, but simply a lack of
invested funds. This underpricing is given by
PM − PD = (
) µA, (7)
therefore it is an increasing function of C/B. It is equal to 0 if
C/B = 0. As the cost of learning tends to zero, more and more funds
are willing to raise money in state D. This brings prices in state
D closer to fundamental value.
This remark extends the model of Shleifer and Vishny (1997) to a
full contracting frame- work, where investors are also able to
commit funding in the low state of nature, i.e., when the asset is
underpriced. What we show is that, when assets are underpriced,
there is an incentive for another class of funds to specialize in
this state. Yet, because arbitrage in this state is costly
(managers need to learn enough about the asset), state D prices
cannot increase too much.
A related point is that learning is necessary in our model to
obtain underpricing. Assume that C is so large that it is never
optimal to learn in period 1. In this case, we are looking for an
equilibrium where both funds investing in states M , and in state
D, make no learning effort. It is easy to verify that there is no
price distortion. The intuition is that entering state D is now
costless as it entails no learning effort: free entry in the two
states eliminates the relative underpricing in state D.
2.3 Predictions of the Model
Our first prediction is related to net of fee conditional returns.
For both types of funds, expected returns conditional on good
performance in period 2 (i.e., in state U) are equal to zero, since
PU = V . This comes from the fact that there is no agency friction
B in this state. Including one would not change the results: both
types of funds would still have the same expected returns in this
state, since they would purchase the same asset at the same
price.
Our model has more interesting predictions on returns conditional
on low performance in period 2:
E(R3|R2 is low, ID > 0) = µ(V −B)− PD = −µαA λD
, (8)
E(R3|R2 is low, ID = 0) = µ (V −B)− PM = −µ(1− α)A λM
. (9)
It appears clearly that expected returns of high effort funds are
larger than expected returns of low effort funds, since high effort
funds invest in state D, where assets are significantly underpriced
(PD < PM ). What is interesting is that this prediction holds in
equilibrium, even though “entry” in both states of nature is free
at the contracting stage.
11
Implication 1. High eort funds exhibit more mean reversion in
returns, in particular when past returns are low:
1. Conditional on high past returns, both funds have similar
expected returns
E(R3|R2 is high, ID > 0) = E(R3|R2 is high, ID = 0).
2. Conditional on low past returns, high eort funds overperform low
eort ones
E(R3|R2 is low, ID > 0) > E(R3|R2 is low, ID = 0).
Before we proceed, we note that the model would have exactly the
same properties if learn- ing effort was contractible. The only
difference is that the incentive compatibility condition on µ would
be replaced by another (slightly weaker) condition on µ to ensure
that high learning effort is optimal for funds invested in state D.
Thus, the differential mean reversion does not hinge on the
contractibility or incontractability of learning effort.
The above prediction is related to Aragon (2007) and Agarwal et al.
(2008), who find empirically that hedge funds with impediments to
withdrawal tend to exhibit superior perfor- mance, even after
controlling for usual risk factors. They interpret this correlation
as evidence that investors demand a premium for holding illiquid
(i.e., locked up) shares.4 And indeed, given the loss of (put)
option value, the cost of illiquidity to investors can be quite
sizeable (Ang and Bollen, 2008, perform a calibration using a real
option model). Our model has the feature that high effort funds may
exhibit higher performance under some circumstances. Using (8) and
(9), we find easily that the excess unconditional performance of
high effort funds is given by
E(R3|ID > 0)− E(R3|ID = 0) = (1− 2α)µA
High effort funds outperform in our model as long as α < 1/2. If
α is small enough, fewer funds invest in state D while more funds
invest in state M . Thus, underpricing in state D is large enough
to make high effort funds outperform low effort ones.
Our model does not only predict that high effort funds’ returns
mean revert more, but also makes predictions on the relative size
of the mean reversion. Intuitively, as the share of high effort
funds α increases, there is more investment in state D, which
increases PD and reduces PM . Given that high effort funds invest
in state D, their outperformance should therefore be reduced. Thus,
across asset markets, α and the overperformance of high effort
funds conditional on past low returns should be negatively
correlated. Given that α and the extent of mean reversion are both
endogenous, we need to verify that this intuition holds in the
model. We do this in the proposition below.
Implication 2. When C decreases: 1. α increases.
4It is interesting to note that the presence of impediments to
withdrawals does not necessarily mean that
investment is illiquid. One possibility is that shares, even though
not immediately redeemable, can be traded
among investors on a secondary market. See for instance Ramadorai
(2008) for a description of such a market.
12
2. The outperformance of high eort funds, conditional on bad
performance,
ρ = E(R3|R2 is low, ID > 0)− E(R3|R2 is low, ID = 0),
decreases.
ρ = (
− α
λD
) µA
is a decreasing function of α. As C decreases, α increases, which
reduces ρ.
We now turn to formal tests of our empirical predictions. We do not
observe learning effort in the data, but we know from the model
that learning effort is high for funds who still receive funding in
state D, i.e., when past performance has been relatively poor. We
use the fact that, in our data, funds with impediments to
withdrawal face lower reductions in assets under management
conditional on bad performance (see also Ding et al., 2008, for
related evidence). Thus, we use the presence as strong impediments
to withdrawal as our measure that ID > 0, and test the two
predictions derived here.
3 Empirical Evidence
3.1 Data Description
We start from a June 2008 download of EurekaHedge, a hedge fund
data provider. The download provides us with monthly data from June
1987 until June 2008. 6,070 funds are initially present in the
sample, with a total of 366,728 observations. Every month, each
fund reports asset under management and net of fee returns. We
delete from the data set all funds that have less than $20m under
management.
Our main results use annual data, but we also use higher frequency
information on returns (monthly and quarterly, see below).
Descriptive statistics on returns and AUM are provided at the
annual frequency in Table 1, Panel A. Mean annual return is about
11% net of fees. Mean assets under management are 300 million
dollars. Also available from the data are fund level
characteristics that do not change over time, whose descriptive
statistics are reported in Panel B. Using these information, we
construct two dummy variables to capture the presence of “strong”
impediments to withdrawal:
• Lockup dummy: In some cases, investors agree to lock their
investment in the fund for a given period of time after their
investment. Out of 5,154 funds for which share restrictions are
known, the mean lockup period is about 2.6 months. This mean
conceals a lumpy distribution: 21% of the funds have lockup
periods, 15% have a lockup period of 12 months, and only 2% have a
longer lockup period. The percentage of funds with lockup periods
that we have in our dataset is similar to what Aragon (2007) has in
his TASS extract.
13
• Redemption dummy: Once the lockup period has passed, investors
can redeem their shares, but still face constraints. Redemption can
only occur at fixed moments of the year. For 3,152 funds (53% of
the total), redemption is monthly. It is quarterly for 25% of the
funds (1,499), and annual in 151 cases. In addition, investors have
to notify the fund of their withdrawal before the redemption
period. This notice period is lower than 1 month in 30% of the
cases, equal to 1 month in 30% of the cases, and is equal or above
one quarter in 15% of the cases. We construct a dummy variable
equal to one when the sum of the redemption and notice periods is
equal or longer than a quarter (90 days). The mean value of this
sum is equal to 92 days; for 38% of the funds, it is equal or
larger than a quarter.
[table 1 about here]
Finally, the spearman correlation between the lockup dummy and the
redemption dummy is 41% (using one data point per fund). Thus, even
though this correlation is positive and statistically significant,
which indicates some complementarity between the two forms of share
restriction, it is far from being equal to 1. In particular, 23% of
the funds without lockup have “redemption periods”.
How effectively constrained are hedge fund investors? To answer
this question, we compute the mean duration of capital, for each
fund, separately for each year. We do this by including the effects
of lockup periods, redemption date and advance notice. We use the
following formula:
Durationit = Noticei + Redemption Periodi
Lockup Periodi∑ s=0
Net Inflowit−s × 1{Net Inflowit−s>0} × (Lockup Periodi − s)
.
The first part of this formula accounts for the effect of notice
and redemption periods. The implicit assumption behind this formula
is that fund’s distance to the next redemption period is uniformly
distributed. The second part accounts for the effect of lockup
periods. For each past net inflow into the fund, it computes the
remaining lockup duration (for instance, 5 month old inflows have a
duration of 7 months if the lockup period is one year). We then
normalize by current assets under management. We use monthly data.
Following the literature on fund flows (Chevalier and Ellison,
1997, Sirri and Tufano, 1998), we compute net inflows by taking the
difference between monthly AUM growth and monthly returns, and
remove outliers. Overall, the above formula is an approximation.
First, past inflows are computed net of outflows. This procedure
leads us to underestimate gross inflows if they occur at the same
time as gross outflows. Second, when shares are still locked up,
the notice and redemption periods are in part ineffective. This
leads the above formula to overestimate duration.
[Figure 1 about here]
14
We plot the sample distribution of estimated durations in Figure 1.
Taking all fund-months in the sample, the sample mean of this
measure is 3 months. On average, the contributions of potential
lockup periods and redemption and notice are of similar sizes. The
25th, 50th and 75th percentiles of the distribution are
respectively 1, 1.5 and 3.5 months. The time series of the mean
duration exhibits a clear downward trend, from 3.5 months in 1996
to about 2.6 months in 2007. If we focus on the subgroup of funds
with lockup periods (21% of our sample), mean duration is,
unsurprisingly, much larger: 8.2 months (median is 5.8). Thus even
though most funds have relatively short duration of liabilities,
there is a group of funds for which the mean dollar of AUM is
secured for at least half a year.
As expected, such impediments to withdrawals do indeed prevent
outflows from happening in the data. To check this, we run the
following regression on annual data:
Outflowit = γi + β.1{ rit−1<r
rf t−1
} × Impedimenti + εit
where Outflowit is a variable equal to 0 if the fund experiences
net inflows in year t, and equal to net inflows if net inflows are
negative. 1{rit−1<r
rf t−1}
is a dummy variable equal to 1 if past year’s returns have been
lower than the risk-free rate, as measured by the yield on 3-month
Treasury Bill. Impedimenti is one of the two measures described
above: a lockup period, or a redemption period of at least a
quarter. We include a fund specific fixed effect γi and cluster
error terms at the year level.
[Table 2 about here]
Regression results are reported in Table 2. As shown in the first
column, if past perfor- mance is below the safe rate of return,
outflows increase by 11% of AUM on average. This is sizeable,
compared to mean annual outflows of 13% in the data, and a cross
sectional stan- dard deviation of 22% (see Table 1). As shown in
columns 2 and 3, such a large sensitivity is somewhat reduced, yet
not totally erased, by the presence of impediments to withdrawal.
Conditional on low performance, funds with such share restrictions
experience outflows of 8% of AUM, compared to 12% without such
restrictions. Hence, the sensitivity is reduced by about one
third.
3.2 Evidence from Conditional Returns
We test here our prediction 1: illiquid funds overperform liquid
funds relatively more after bad performance than after good
performance. To do this, we first run the following
regression:
rit = γi + β.1{ rit−1<r
rf t−1
rf t−1
} × Impedimenti + εit (10)
where rit is the annual return of fund i in year t. We use annual
data because at the annual frequency returns are less likely to be
polluted by asset illiquidity problems (Lo, 2008; more on this
below). γi is a fund-specific fixed effect, designed to capture
heterogeneity in risk exposure and alphas, across funds (but our
results are unchanged in the absence of fixed effects). We cluster
error terms at the year level. Our theory predicts that the extent
of mean reversion in returns should be larger for illiquid funds,
i.e., δ > 0.
15
[Table 3 about here]
Table 3 reports the results. Consistently with the first prediction
of our model, the mean reversion of returns significantly increases
with impediments to withdrawal. After returns below the risk-free
rate, annual returns increase by 2.6 points for funds with no
lockup and by 7.7 points for funds with a lockup; the difference is
strongly significant (column 1). When we look at redemption
periods, we find that returns increase by 3.4 points after low
performance when the notice + redemption period is shorter than a
quarter, and by 6.7 when it is longer than a quarter (column 2).
Again, the difference is strongly significant.
Individual fixed effects may introduce a mechanical tendency to
generate mean reversion of returns in the regressions. Indeed,
adding a fund specific fixed effect amounts to replacing every
variable by its difference from the individual average. If a fund’s
return in year t − 1 is below its average return, then its returns
in year t will tend to be above average. The bias may be more
severe, the smaller the number of time periods. This is a priori
not a problem for the regressions in columns 1 and 2, since we are
interesting in the difference in mean reversion between funds with
a lockup and funds with no lockup, not in the absolute level of
mean reversion. In terms of equation (10), the estimate of β might
be biased, but the estimate of δ should not. In particular, funds
with lockups appear in the data during approximatively the same
length of time (4.5 years on average) as funds with no lockup (4.6
years on average). To confirm that intuition, we rerun our
regressions without fixed effects. Consistently with a bias in the
absolute level of mean reversion when fund specific fixed effects
are used, the estimates of β are significantly lower without fixed
effects (columns 3 and 4). Furthermore, the difference in mean
reversion between funds with a lockup and funds with no lockup is
not modified and remains strongly significant. Results are similar
when impediments to withdrawal are measured with the quarterly
redemption dummy.
Note that each year an average of 6% of funds drop from the data.
This could potentially bias our results if exit is correlated both
with past returns and with current (at the time of exit) returns,
and if these correlations are different for funds with a lockup and
funds with no lockup. Funds can drop from the data either because
they are liquidated, or because they voluntarily stop reporting
their returns. The former is presumably associated with poor
performance. The latter might occur when the fund is doing well.
Indeed, funds report to data vendors for the purpose of indirect
marketing to potential investors; when a fund performs well and has
large capital flows, it may have no incentive to continue
reporting. If, for instance, funds exit the database after bad
performance, and also perform poorly the year of liquidation, but
the econometrician does not observe that last poor performance,
then our regressions overestimate the mean reversion of returns.
However, there is a priori no reason to believe that this bias
should be correlated with impediments to withdrawal, hence the test
of our prediction 1 should not be biased. To check further that
intuition, we construct a dummy variable equal to one the first
year a fund disappears from the data. We then estimate a Logit
model to explain the exit dummy with our variable of bad
performance interacted with our measures of impediment to
withdrawal.
[Table 4 about here]
16
Results are reported in Table 4. We find that exit is about twice
more likely after a performance below the risk-free rate, and that
this effect is significant. We also find that funds with a lockup
are about 30% less likely to exit the database, and funds with a
long redemption period about 25% less likely. Again, these effects
are significant. However, we find no significant evidence that the
relation between exit and poor performance is affected by
impediments to withdrawal. If any, quarterly redemptions reduce the
probability of exit after poor performance by about 30%, but this
effect is statistically insignificant. Overall, these results
suggest that, if there is any bias due to exit, the bias tends to
underestimate the relative mean reversion of illiquid funds. We
check in unreported regressions that if a poor performance is
assigned to funds that exit the database, then our results are
reinforced.
Results from Table 3 show that there is a significantly larger
tendency for returns of illiquid funds to mean revert, but it does
not differentiate between mean reversion in bad states of nature
and mean reversion in good states of nature. Our model, however,
does predict such an asymmetry. Prediction 1 suggest that most of
the mean reversion should be conditional on bad states of nature.
This comes from the fact that, in the model, bad states of nature
are states where assets are underpriced, while there is no
mispricing in the good states of nature. Such a prediction holds
even if µ = 1, i.e., expected asset payoffs are similar in states U
, M , and D.
To test it, we check if there is a difference in mean reversion
between funds with past low returns, and events with past high
returns. We define low returns as above, i.e., as cases when
returns are below the safe rate of return. This corresponds to
approximately 21% of the observations in our dataset. Against this
background, we define high returns as cases where past annual
returns are above 20% net of fee: this threshold is chosen because
it isolates about 21% of our fund-year observations, so the
identifying power of the data should be similar for mean reversion
conditional on high and low returns.
[Table 5 about here]
We then run regressions similar in spirit to (10): we also
introduce a dummy variable equal to one if past returns have been
above 20%, in addition to the dummy that past returns are below the
risk-free rate. Results are reported in Table 5. In column 1, we
interact these two terms with the lockup dummy. The asymmetry
appears clearly: conditional on bad performance, funds with lockups
overperform funds without lockups by about 5 percentage points.
Conditional on good past performance, both categories of funds
experience the same decline in expected returns. In column 2, the
same asymmetry is present with our second measure of impediment to
withdrawal. In columns 3 and 4, we rerun these two regressions
without the fund specific fixed effects. The asymmetry between bad
states of nature and good states of nature continue to appear
significantly.
3.3 Mean Reversion Conditional on Other Funds Capital
Structures
In this section, we test our prediction 2. This prediction states
that the difference between high and low effort funds’ expected
returns, conditional on low past performance, is a decreasing
17
function of α. When the fraction of funds with ID > 0 decreases,
the price difference between states D and M will expand, which will
increase the scope for mean reversion for funds that invest in
state D. In our model, α is determined in equilibrium but the proof
of prediction 2 ensures that this negative covariation should be
there as underlying parameters change.
Econometrically, prediction 2 rewrites, for fund i at date t
operating on asset market s
rist = γi + β.1{ rist−1<r
rf t−1
rf t−1
} × Impedimenti × αs + εit (11)
where αs corresponds to α in the model; αs is the fraction of other
illiquid funds. Ideally, to compute αs, we should focus on pure
players on the same asset market where fund i
operates. It is, however, not feasible, first because in general
funds may operate on several assets markets, and also because the
data does not provide us with the positions of funds on each
market. Thus, we use the style classification of the database,
implicitly assuming that funds operating within a given style buy
the same assets. Thus, αs will be the fraction, in the style where
fund i is operating, of funds with a lockup, or the fraction of
funds with at least a quarterly redemption. Our model unambiguously
predicts that the difference in mean reversion should be lower in
styles where there are more other funds with impediments to
withdrawal, so that η < 0.
[table 6 about here]
Table 6 reports estimates of equation (11) on our sample. Column 1
measures impediments as lockups. Conditional on past low
performance, illiquid funds outperform by about 13 percentage
points if there no other illiquid funds at all in the style. If the
fraction of other funds with lockups is 20% (the sample average),
the outperformance of illiquid funds goes down to 7 percentage
points. The differential effect of αs is statistically significant
at the 5% level. In column 2, we use quarterly redemptions as our
measure of impediments to withdrawals, both at the fund level and
to compute αs. With this specification, the result is less
favorable to our theory: the estimate of η is insignificant and
economically negligible.
3.4 Comparison with Existing Literature
Our main empirical result is thus that funds experience more mean
reversion in returns when they have share restrictions. In this
section, we show how our results complement the existing literature
on hedge funds.
Getmansky, Lo and Makarov (2004) have designed a measure of returns
“smoothing” by funds (named θ0). This measure is econometrically
complex to put in place, but the principle is to look at
autocorrelation of monthly returns. If monthly returns are very
autocorrelated, then it is likely that funds smooth returns across
months to minimize volatility. Such a strategy is easier to put in
place for assets whose prices cannot easily marked to market, so
θ0
is also considered as a proxy for asset illiquidity. Consistent
with the idea that impediments
18
to withdrawals help funds to buy illiquid assets, Aragon (2007),
Ding et al. (2008) and Liang and Park (2008) have shown that high
θ0 funds also tend to have share restrictions.
This contradiction with our results (we find less returns
persistence for funds with share restrictions) is only apparent,
because we focus on annual returns, which are less likely to be
smoothed. Illiquidity and window dressing issues should therefore
generate less autocorrela- tion at this frequency. In fact, as we
show in Tables 3 and 4, column 1, annual returns are more likely to
mean revert, for the average hedge fund in our sample.
This difference between existing results and ours does not come
from the fact that we are using a different dataset (most papers
use Lipper/TASS), but really from the fact that we work with annual
data. To check this, we look at the correlation between impediments
to withdrawal and the autocorrelation of monthly returns. First, we
compute, for each fund, a measure of the first order
autocorrelation in monthly returns: we find an average of 0.10.
This figure is consistent with what can be found in papers using
other datasets: for instance, Lo (2008) finds that the mean first
order autocorrelation of fund returns is 0.08 using the TASS/Lipper
database (his Table 2.6). Consistently with Liang and Park (2008),
we also find that autocorrelation is positively correlated with the
presence of a lockup period: in our dataset, the correlation is
0.08 (compared to 0.09 in their study), statistically significant
at the 1% level. Thus, our data generates the same pattern as
existing papers: impediments to withdrawal are associated with more
autocorrelated monthly returns.
[Table 7 about here]
To further check this, we run regression (10) with monthly, instead
of yearly, data, on all funds. We find, in the first column of
Panel A of Table 7, that coefficient on low past performance is
−0.38 and the interaction term is −0.14, but significant at the 10%
level only. Thus, at very high frequency, our data, like others,
generate the positive autocorrelation pattern found in the
literature.
This analysis suggests that the correlation between impediments to
withdrawals and au- tocorrelation of returns is affected by two
opposite forces: at the annual frequency, illiquid funds mean
revert more because they take advantage of temporary mispricing,
while at the monthly frequency, they exhibit more autocorrelation
because they hold illiquid assets whose prices display a
significant inertia. To disentangle these two effects, we run
regression (10) on different groups of hedge funds, depending on
the liquidity of the asset market they operate on. If we focus on
funds managing liquid assets such as (long-short) equity funds, we
find some evidence that persistence of monthly returns is weaker
for funds with lockup periods (Panel A, column 2), although the
relation is not statistically significant. So even with monthly
data, the evidence from “liquid styles” is more in line with our
theory. For fixed income funds, the pattern is reversed (Panel A,
column 3): for this strategy, as the existing literature would
predict, share restrictions means more smoothing, and therefore
more autocorrelation. In unreported regressions, we find similar
results when we measure impediments to withdrawal with redemption
periods.
With quarterly data, our predictions start to have more bite. In
Panel B of Table 7, we re-estimate equation (10), using as the LHS
variable quarterly, instead of monthly, returns.
19
On the whole panel of funds, we find evidence of more mean
reversion for funds with lockup periods (Panel B, column 1), while
there was less mean reversion at the monthly frequency (Panel A,
column 1). For equity funds, the effect is even more spectacular
(Panel B, column 2). For fixed income funds, the movement of
persistence is still present at the quarterly frequency, but less
significant (Panel B, column 3). In unreported regressions, we find
similar results when we measure impediments to withdrawal with
redemption periods.
4 Conclusion
In this paper, we have developed and tested a model of delegated
fund management in equi- librium. The starting point was Shleifer
and Vishny (1997): arbitrageur invest in a common market. Arbitrage
opportunities may generate temporary underperformance. Our model
ex- plicitly models the contract that ties the investor and the
fund manager. Because in some cases the underpricing can be so
severe, we find that it is always optimal for the investor to
commit not to redeem his shares, for some funds only. Hence, we
predict that funds with share restrictions will outperform those
without such restrictions after past bad performance. We find
evidence consistent with this in the data.
20
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22
All funds with AUM>20m Mean Std Obs
Panel A: Annual variables Return 10.5 13.5 7,041 AUM 301 554 7,041
Net Inflows 18.0 75.3 7,041 Net Inflows× (Net Inflows<0) -13.0
22.6 7,041
Panel B: Fixed Characteristics Long Short 0.44 - 5,310 Global Macro
0.06 - 5,310 Fixed Income 0.05 - 5,310 Lockup Period (months) 2.6
5.8 5,154 Lockup dummy 0.21 - 5,154 Notice + Redemption Periods 92
84 4,805 Quarterly Not.+Red. dummy 0.38 - 4,805
Data: EurekaHedge, 1994-2007. Annual data, excluding funds with AUM
lower than 20 million USD.
23
Table 2: Outflows and Impediments to Withdrawal
Dependent variable Net inflowsit × (Net inflowsit < 0)
Impediment to withdrawal None Lockup Quarterly
Redemption (1) (2) (3)
(0.02) (0.02) (0.02) (rit−1 < rrf
t−1) - 0.04∗∗ 0.04∗∗∗
× Impedimenti (0.02) (0.01)
Fund FE Yes Yes Yes Observations 4,825 4,690 4,162 Adj. R2 0.54
0.54 0.55
Data: EurekaHedge, 1994-2007. Annual data, excluding funds with AUM
lower than 20 million USD. The
dependent variable is equal to annual net inflows if they are
negative, and zero else. Net inflows are computed
as the difference between the growth in AUM minus net-of-fee
returns. All specifications include fund specific
fixed effects. In column (1), the only regressor is a dummy equal
to 1 if the past annual return was lower than
the yield on the 3 month T-bill. In column (2), we interact with
the fact that fund i has a lockup period of
at least a year. In column (3) we interact with the fact that
redemption + notice periods is at least 120 days.
Error terms are clustered at the year level. ∗, ∗∗, and ∗∗∗ means
statistically different from zero at 10, 5 and
1% levels of significance.
Dependent variable rit Impediment to withdrawal Lockup Quarterly
Lockup Quarterly
Redemption Redemption (1) (2) (3) (4)
Impedimenti - - -0.7 1.6 (0.8) (1.1)
(rit−1 < rrf t−1) 2.6 3.4 -3.4 -3.2∗
(2.3) (2.0) (2.1) (1.8) (rit−1 < rrf
t−1) 5.1∗∗∗ 3.3∗∗∗ 5.4∗∗∗ 5.4∗∗∗
× Impedimenti (1.4) (1.0) (1.8) (1.3)
Fund FE Yes Yes No No Observations 4,412 3,902 4,412 3,902 Adj. R2
0.48 0.49 0.01 0.02
Data: EurekaHedge, 1994-2007. Annual data, excluding funds with AUM
lower than 20 million USD. The
dependent variable is the annual net-of-fee return. In columns (1)
and (2) the specifications include fund
specific fixed effects; in columns (3) and (4) they do not. In
columns (1) and (3), the regressors are a dummy
equal to 1 if the past annual return was lower than the yield on
the 3-month T-bill, and that dummy interacted
with the fact that fund i has a lockup. In columns (2) and (4), we
interact with the fact that redemption +
notice periods is at least 90 days. Error terms are clustered at
the year level. ∗, ∗∗, and ∗∗∗ means statistically
different from zero at 10, 5 and 1% levels of significance.
25
Dependent variable Exitit
(1) (2) Impedimenti -0.3∗ -0.2∗∗
(0.2) (0.1) (rit−1 < rrf
t−1) 1.0∗∗∗ 1.1∗∗∗
t−1) -0.1 -0.3 × Impedimenti (0.3) (0.3)
Observations 4,707 4,171 Pseudo R2 0.03 0.03
Data: EurekaHedge, 1998-2007. Annual data, excluding funds with AUM
lower than 20 million USD. In all
the specifications, a Logit model is estimated in which the
dependent variable is a dummy equal to one if the
fund exits from the data in the current year. In columns (1), the
regressors are a dummy equal to 1 if the past
annual return was lower than the yield on the 3-month T-bill, and
that dummy interacted with the fact that
fund i has a lockup. In columns (2), we interact with the fact that
redemption + notice periods is at least 90
days. Error terms are clustered at the year level. ∗, ∗∗, and ∗∗∗
means statistically different from zero at 10, 5
and 1% levels of significance.
26
Table 5: Conditional Returns and Impediments to Withdrawal: Good
States versus Bad States
Dependent variable rit Impediment to withdrawal Lockup Quarterly
Lockup Quarterly
Redemption Redemption (1) (2) (3) (4)
Impedimenti - - -0.3 1.6 (0.7) (0.9)
(rit−1 < rrf t−1) 1.8 2.8 -1.7 -1.7
(1.9) (1.7) (2.0) (1.7) (rit−1 > 20%) -2.9∗ -2.1 6.3∗∗∗
6.3∗∗
(1.5) (2.0) (2.1) (2.8) (rit−1 < rrf
t−1) 5.0∗∗∗ 2.8∗∗∗ 5.0∗∗∗ 5.4∗∗∗
× Impedimenti (1.4) (0.8) (1.5) (1.0) (rit−1 > 20%) 0.5 -1.0
-1.5 -1.1 × Impedimenti (1.0) (2.1) (1.9) (2.1)
Fund FE Yes Yes No No Observations 4,412 3,902 4,412 3,902 Adj. R2
0.48 0.49 0.04 0.04
Data: EurekaHedge, 1994-2007. Annual data, excluding funds with AUM
lower than 20 million USD. The
dependent variable is the annual net-of-fee return. In columns (1)
and (2) the specifications include fund
specific fixed effects; in columns (3) and (4) they do not. In
columns (1) and (3), the regressors are a dummy
equal to 1 if the past annual return was lower than the yield on
the 3-month T-bill, a dummy equal to 1 if
the past annual return was above 20%, and these dummies interacted
with the fact that fund i has a lockup.
In columns (2) and (4), we interact with the fact that redemption +
notice periods is at least 90 days. Error
terms are clustered at the year level. ∗, ∗∗, and ∗∗∗ means
statistically different from zero at 10, 5 and 1%
levels of significance.
Table 6: Mean Reversion: The Impact of Other Funds’
Illiquidity
Dependent variable rit Impediment to withdrawal Lockup
Quarterly
Redemption (1) (2)
(2.4) (2.0) (rit−1 < rrf
t−1) 12.7∗∗∗ 3.8 × Impedimenti (3.3) (5.5) (rit−1 < rrf
t−1) 13.7∗∗ -4.4 × αst (5.1) (3.3) (rit−1 < rrf
t−1) -31.1∗∗ 0.3 × Impedimenti × αst (13.8) (12.0)
Fund FE Yes Yes Observations 4,412 3,902 Adj. R2 0.49 0.49
Data: EurekaHedge, 1994-2007. Annual data, excluding funds with AUM
lower than 20 million USD. The
dependent variable is the annual net-of-fee return. All
specifications include fund specific fixed effects. In
column (1), we use as measure of impediment to withdrawal the fact
that fund i has a lockup. In column
(2) the impediment dummy is equal to 1 if redemption + notice
periods is at least 90 days. Error terms are
clustered at the year level. ∗, ∗∗, and ∗∗∗ means statistically
different from zero at 10, 5 and 1% levels of
significance.
28
Table 7: Conditional Returns and Impediments to Withdrawal: Higher
Frequency Evidence
Dep. Variable rit Panel A: Monthly frequency
All Long short equity Fixed Income (1) (2) (3)
(rit−1 < rrf t−1) -0.38∗∗ -0.44∗∗ -0.45∗∗∗
(0.18) (0.21) (0.14) (rit−1 < rrf
t−1) -0.14∗ 0.14 -0.49∗∗∗
× Lockupi (0.08) (0.13) (0.14)
Fund FE Yes Yes Yes Observations 120,734 51,963 6,929 Adj. R2 0.06
0.05 0.10
Panel B: Quarterly frequency All Long short equity Fixed Income (1)
(2) (3)
(rit−1 < rrf t−1) -0.10 -0.42 -0.49
(0.42) (0.58) (0.40) (rit−1 < rrf
t−1) 0.53∗∗ 1.37∗∗∗ -1.24∗∗
× Lockupi (0.26) (0.24) (0.55)
Fund FE Yes Yes Yes Observations 34,447 14,828 1,989 Adj. R2 0.15
0.15 0.18
Data: EurekaHedge, 1994-2007. Annual data, excluding funds with AUM
lower than 20 million USD. The
dependent variable is the net-of-fee return. Panel A uses monthly
returns, while Panel B uses quarterly returns.
All specifications include fund specific fixed effects. In column
(1), we look at all funds. In column (2), we
restrict the sample to the long-short equity style. In column (3),
we restrict the sample to funds operating in
the fixed income style. Error terms are clustered at the month
(Panel A) and quarterly (Panel B) level. ∗, ∗∗,
and ∗∗∗ means statistically different from zero at 10, 5 and 1%
levels of significance.
29
Test
Duration of Capital: Descriptive Statistics Mean earliest possible
withdrawal of AUM = notice + (redemption/2) + past inows x
remaining period under lock-up
Johan Hombert, David Thesmar (ENSAE-CREST, HEC&CEPR)Limits of
Limits of Arbitrage Theory and Evidence March 6, 2009 15 / 24Data:
EurekaHedge, 1994-2007. Monthly data, excluding funds with AUM
lower than 20 million USD.
30